The ampleness of canonical sheaves and the proof of "$X \simeq \mathrm{Proj}\left(\bigoplus_k H^0(X, \omega_X^k)\right)$".

Question

In Bondal and Orlov's paper, ''Reconstruction of a variety from the derived category and groups of autoequivalences'' (http://www.mi-ras.ru/~orlov/papers/Compositio2001.pdf), I think that they use the following result:

Let X be the smooth projective variety, $\omega_X$ is the canonical sheaf of $X$. If $\omega_X$ is ample,
$$X \simeq\mathrm{Proj}\left(\bigoplus_k H^0(X, \omega_X^k)\right). $$

I am looking for a complete proof of this fact. I have read some literature, but the proof is omitted in them. Is this a simple fact? Could you tell me the proof or the literature on which it is listed?